L(s) = 1 | + (−1.22 + 0.707i)2-s + (−2.24 + 1.98i)3-s + (0.999 − 1.73i)4-s + (−5.32 + 3.07i)5-s + (1.34 − 4.02i)6-s + (−4.69 + 5.19i)7-s + 2.82i·8-s + (1.09 − 8.93i)9-s + (4.34 − 7.52i)10-s + (15.5 + 8.99i)11-s + (1.19 + 5.87i)12-s + 1.38·13-s + (2.07 − 9.68i)14-s + (5.84 − 17.4i)15-s + (−2.00 − 3.46i)16-s + (−5.32 − 3.07i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.748 + 0.662i)3-s + (0.249 − 0.433i)4-s + (−1.06 + 0.614i)5-s + (0.224 − 0.670i)6-s + (−0.670 + 0.742i)7-s + 0.353i·8-s + (0.121 − 0.992i)9-s + (0.434 − 0.752i)10-s + (1.41 + 0.818i)11-s + (0.0998 + 0.489i)12-s + 0.106·13-s + (0.147 − 0.691i)14-s + (0.389 − 1.16i)15-s + (−0.125 − 0.216i)16-s + (−0.313 − 0.180i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.131723 + 0.428326i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.131723 + 0.428326i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 - 0.707i)T \) |
| 3 | \( 1 + (2.24 - 1.98i)T \) |
| 7 | \( 1 + (4.69 - 5.19i)T \) |
good | 5 | \( 1 + (5.32 - 3.07i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-15.5 - 8.99i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 1.38T + 169T^{2} \) |
| 17 | \( 1 + (5.32 + 3.07i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-7.53 - 13.0i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (22.9 - 13.2i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 19.0iT - 841T^{2} \) |
| 31 | \( 1 + (9.22 - 15.9i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 11.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 54.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-46.4 + 26.8i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-35.4 - 20.4i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (47.2 + 27.2i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-35.8 - 62.1i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (10.9 - 19.0i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 109. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-7.11 + 12.3i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (46.3 + 80.3i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 35.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-138. + 80.1i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 118.T + 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.10796176558371094793387268974, −15.46153385315216259525495636915, −14.54785280924616020048397116816, −12.14406439962143941490376377698, −11.54971465356049974863298205224, −10.09216547197202675684866194062, −9.061066157473784969236716959127, −7.22693481106945414293389386481, −6.02115894214775578058465410881, −3.91588356256673611795640862293,
0.67547929323321649937980912680, 4.02915247494830154081848320648, 6.43221073517784398833118934603, 7.67871547714185533327509634551, 9.028532190082308639088079466301, 10.76923232768022871449356971956, 11.73052378817740974278149445531, 12.58964691020056133798681408978, 13.86673303648791370518892775739, 15.96506674573321790706084099769